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Metamath Proof Explorer

Theorem imcj

Description: Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005) (Revised by Mario Carneiro, 14-Jul-2014)

Ref Expression
Assertion imcj ( 𝐴 ∈ ℂ → ( ℑ ‘ ( ∗ ‘ 𝐴 ) ) = - ( ℑ ‘ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 recl ⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℝ )
2 1 recnd ⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℂ )
3 ax-icn ⊢ i ∈ ℂ
4 imcl ⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℝ )
5 4 recnd ⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℂ )
6 mulcl ⊢ ( ( i ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) ∈ ℂ ) → ( i · ( ℑ ‘ 𝐴 ) ) ∈ ℂ )
7 3 5 6 sylancr ⊢ ( 𝐴 ∈ ℂ → ( i · ( ℑ ‘ 𝐴 ) ) ∈ ℂ )
8 2 7 negsubd ⊢ ( 𝐴 ∈ ℂ → ( ( ℜ ‘ 𝐴 ) + - ( i · ( ℑ ‘ 𝐴 ) ) ) = ( ( ℜ ‘ 𝐴 ) − ( i · ( ℑ ‘ 𝐴 ) ) ) )
9 mulneg2 ⊢ ( ( i ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) ∈ ℂ ) → ( i · - ( ℑ ‘ 𝐴 ) ) = - ( i · ( ℑ ‘ 𝐴 ) ) )
10 3 5 9 sylancr ⊢ ( 𝐴 ∈ ℂ → ( i · - ( ℑ ‘ 𝐴 ) ) = - ( i · ( ℑ ‘ 𝐴 ) ) )
11 10 oveq2d ⊢ ( 𝐴 ∈ ℂ → ( ( ℜ ‘ 𝐴 ) + ( i · - ( ℑ ‘ 𝐴 ) ) ) = ( ( ℜ ‘ 𝐴 ) + - ( i · ( ℑ ‘ 𝐴 ) ) ) )
12 remim ⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ 𝐴 ) = ( ( ℜ ‘ 𝐴 ) − ( i · ( ℑ ‘ 𝐴 ) ) ) )
13 8 11 12 3eqtr4rd ⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ 𝐴 ) = ( ( ℜ ‘ 𝐴 ) + ( i · - ( ℑ ‘ 𝐴 ) ) ) )
14 13 fveq2d ⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ ( ∗ ‘ 𝐴 ) ) = ( ℑ ‘ ( ( ℜ ‘ 𝐴 ) + ( i · - ( ℑ ‘ 𝐴 ) ) ) ) )
15 4 renegcld ⊢ ( 𝐴 ∈ ℂ → - ( ℑ ‘ 𝐴 ) ∈ ℝ )
16 crim ⊢ ( ( ( ℜ ‘ 𝐴 ) ∈ ℝ ∧ - ( ℑ ‘ 𝐴 ) ∈ ℝ ) → ( ℑ ‘ ( ( ℜ ‘ 𝐴 ) + ( i · - ( ℑ ‘ 𝐴 ) ) ) ) = - ( ℑ ‘ 𝐴 ) )
17 1 15 16 syl2anc ⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ ( ( ℜ ‘ 𝐴 ) + ( i · - ( ℑ ‘ 𝐴 ) ) ) ) = - ( ℑ ‘ 𝐴 ) )
18 14 17 eqtrd ⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ ( ∗ ‘ 𝐴 ) ) = - ( ℑ ‘ 𝐴 ) )